Interest Formulas. Simple Interest

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Jeffery Webb
6 years ago
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1 Interest Formulas You have $1000 that you wish to invest in a bank. You are curious how much you will have in your account after 3 years since banks typically give you back some interest. You have several options to choose from, each with a different way that the bank calculates interest. The annual interest rate in each bank is at 10%. The first bank calculates interest using. When a bank calculates interest by simple interest, the calculation is always based off of your principal which is your initial deposited amount. The interest is calculated by multiplying the principal with the annual interest rate. This amount gets added to your account at the end of each year. The amount in your account is given by the Simple Interest Formula, Amount in account at time Principal (Initial amount in dollars) Annual Interest Rate (expressed as a decimal) Time in years If we put in $1000 in this bank with simple interest at 10%, we can determine how much our money will grow by looking at the graph of The amount in our account grows linearly through time. If we want to know how much is in our account after 3 years, we simply evaluate our function at : We will accumulate $300 in interest after 3 years. In fact, every year we will receive in interest. We now look at the second bank that introduces what is called compounded interest.

2 Compounding Interest Compounding interest calculates interest from our principal as well as our accumulated interest. This appears to be a better choice since our interest will be calculated from a larger value every year, or compounding interval. In fact, sometimes the interest will be calculated several times a year. We define the number of times that interested is calculated per year, the compounding interval In this example, let s say the bank only calculates interest once a year. Therefore there is one compounding period per year. The amount in your account given by the Compounding Interest Formula is, number of compounding intervals per year If we invest $1000 in this bank that earns 10% compounded yearly, we can determine how much our money will grow by looking at the graph of The amount in our account grows at a faster rate through time because interest is calculated by the accumulated amount in our account. If we want to know how much is in our account after 3 years, we simply evaluate our function at : We will accumulate in interest after years Let s break it down and compare this with simple interest Compounding Interest Interest Amount Interest Amount End of Year 1 $1100 End of Year 2 $1200 $1210 End of Year 3 $1300

3 Compounded interest determines interest from the current account balance, not the principal. What would happen if the number of compounding intervals,, increased? This would mean that interest would be calculated more times per year and we would expect that we would have more money in our account after three years Let s consider our scenario of investing $1000 and see what we would have in our account for various numbers of compounding intervals. Compounding Frequency Annually 1 $ Semiannually 2 $ Quarterly 4 $ Monthly 12 $ Weekly 52 $ Daily 365 $ Hourly 8,760 $ We notice that as we increase our compounding interval,, the amount in our account increases but appears to be approaching a certain value. In fact, if we look at a graph of with respect to we can easily notice this trend: Let s try and determine the value that our bank account will approach when we let approach infinity. take constant outside multiply exponent by Let it property

4 This is where the definition of the constant is helpful in determining the it. By definition, is an irrational number and its value is approximately equal to Let. Therefore, Thus, This is the Continuous Compounding Interest Formula. Continuous Compounding Interest Banks will sometimes calculate interest by continuous compounding. We showed that this is simply the it as the number of compounding intervals goes to infinity. You can think of this as calculating interest every second, but we have a nice form for us to easily determine the amount in our account at any time. If we invest $1000 in this bank that earns 10% compounded continuously, we can determine how much our money will grow by looking at the graph of The amount in our account should grow faster that it did when it was compounded yearly. The amount should also approach the value that we saw it tending to in our table and graph of with respect to. If we want to know how much is in our account after 3 years, we simply evaluate our function at :

5 Which bank should we choose? Obviously when the annual interest rates are the same, continuously compounded interest gives us the most in return. A graph of, Compounding Interest (, and Continuous Compounding Interest will help further drive this point: Continuous Compounding Interest Compound Interest Let s say we chose the bank with continuous compounding interest and we have a financial goal of $4000. We want to know when we will have $4000 in our bank. There are two methods in solving this. Algebraic Method Let be the time when we have reached our financial goal of $4000. We will use the Continuous Compounding Interest Formula to determine : Graphical Method In your calculator, graph the function and a horizontal line indicating the amount level of $4000. Y1 = 1000e^(0.1X) Y2 = 4000 years When you graph them, find the intersection by going to CALC and selecting 5: intersect. You will be asked to select both curves and then guess where the intersection is. If you are having trouble seeing your graph, go to the TABLE to determine appropriate X and Y min and max values. When you find the intersection, you will then get the following coordinates: X= Y=4000 The -coordinate corresponds to the time in years when your account reaches $4000.

Daily Outcomes: I can evaluate, analyze, and graph exponential functions. Why might plotting the data on a graph be helpful in analyzing the data?

3 1 Exponential Functions Daily Outcomes: I can evaluate, analyze, and graph exponential functions Would the increase in water usage mirror the increase in population? Explain. Why might plotting the data